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A000408 Numbers that are the sum of three nonzero squares. 104
3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 27, 29, 30, 33, 34, 35, 36, 38, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102, 104 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n) !== 7 (mod 8). - Boris Putievskiy, May 05 2013
A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015
According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 nonzero squares (i.e., is in A000378 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1,2,5,10,13,25,37,58,85,130,?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - Jeffrey Shallit, Jan 15 2017
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 267.
Savin Réalis, Answer to question 25 ("Toute puissance entière de 3 est une somme de trois carrés premiers avec 3"), Mathesis 1 (1881), pp. 87-88. (See also p. 73 where the question is posed.)
LINKS
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.
S. Mezroui, A. Azizi, and M'hammed Ziane, On a Conjecture of Farhi , J. Int. Seq. 17 (2014) #14.1.8.
FORMULA
a(n) = 6n/5 + O(log n). - Charles R Greathouse IV, Mar 14 2014; error term improved Jul 05 2024
MAPLE
N:= 1000: # to get all terms <= N
S:= series((JacobiTheta3(0, q)-1)^3, q, 1001):
select(t -> coeff(S, q, t)>0, [$1..N]); # Robert Israel, Jan 14 2016
MATHEMATICA
f[n_] := Flatten[Position[Take[Rest[CoefficientList[Sum[x^(i^2), {i, n}]^3, x]], n^2], _?Positive]]; f[11] (* Ray Chandler, Dec 06 2006 *)
pr[n_] := Select[ PowersRepresentations[n, 3, 2], FreeQ[#, 0] &]; Select[ Range[104], pr[#] != {} &] (* Jean-François Alcover, Apr 04 2013 *)
max = 1000; s = (EllipticTheta[3, 0, q] - 1)^3 + O[q]^(max+1); Select[ Range[max], SeriesCoefficient[s, {q, 0, #}] > 0 &] (* Jean-François Alcover, Feb 01 2016, after Robert Israel *)
PROG
(PARI) is(n)=for(x=sqrtint((n-1)\3)+1, sqrtint(n-2), for(y=1, sqrtint(n-x^2-1), if(issquare(n-x^2-y^2), return(1)))); 0 \\ Charles R Greathouse IV, Apr 04 2013
(PARI) is(n)= my(a, b) ; a=1 ; while(a^2+1<n, b=1 ; while(b<=a && a^2+b^2<n, if(issquare(n-a^2-b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0) ;
for(n=3, 1e3, if(is(n), print1(n, ", "))); \\ Altug Alkan, Jan 18 2016
(Haskell)
a000408 n = a000408_list !! (n-1)
a000408_list = filter ((> 0) . a025427) [1..]
-- Reinhard Zumkeller, Feb 26 2015
(Python)
def aupto(lim):
squares = [k*k for k in range(1, int(lim**.5)+2) if k*k <= lim]
sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
sum3sqs = set(a+b for a in sum2sqs for b in squares)
return sorted(set(range(lim+1)) & sum3sqs)
print(aupto(104)) # Michael S. Branicky, Mar 06 2021
CROSSREFS
Sequence in context: A065940 A358350 A024795 * A025321 A153238 A343112
KEYWORD
nonn,changed
AUTHOR
STATUS
approved

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Last modified July 12 23:13 EDT 2024. Contains 374257 sequences. (Running on oeis4.)